Approximate affine linear relationship between L 1 norm objective functional values and L 2 norm constraint bounds

Abstract

For an optimization problem with an norm objective function subject to an norm inequality constraint, this paper shows that there is an approximately linear relationship between the norm objective functional values and the norm specifications. This relationship is verified through the use of random and real world industrial data. The obtained results can be employed for 1) estimating the norm output objective functional value without solving the optimization problem numerically; 2) providing an insight for defining the norm specification in which a simple method is proposed in this paper; and 3) testing whether the obtained solutions are the globally optimal solutions or not. These advantages are demonstrated via the use of random data.